Joint Decision of Pricing and Cross Ruff Coupon Value under Substitute (Complement) Demand

Joint Decision of Pricing and Cross-Ruff Coupon Value

under Substitute (Complement) Demand

Zhongguo Shi, Yinping Mu

School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, China Email: [email protected]

Received October 10th_{, 2011; revised December 9}th_{, 2011; accepted December 26}th_{, 2011}

ABSTRACT

This paper focuses on the optimal decisions of pricing and cross-ruff coupon face value for two linked brands products. We develop the profit maximization models for two kinds of coupons: in-pack cross-ruff coupon and on-pack cross-ruff coupon under substitute (complement) demand, and conclude that 1) the coupon values, prices and profit under substi-tute demand are higher than under complement demand or independent demand; 2) the profit under on-pack coupon mode are higher than in-pack coupon mode. According to the results, we provide the managerial insights for choosing the linked brands. Through numerical example, we observe that the coupon values, target brand prices and carrier brand prices for on-pack cross-ruff coupon are more sensitive than in-pack cross-ruff coupon to reference price.

Keywords: Cross-Ruff Coupon; Carrier Brand; Target Brand; Pricing

1. Introduction

Coupon is a very typical method of promotion widely used by manufacturers and retailers. Coupons are superi- or to direct discount for the commodity price, because di- rect price discount will affect consumers’ price expecta- tions or form price discrimination in the market. Accor- ding to the report issued by Marketing Services Company of NCH Group, coupon is the most powerful method to stimulate demand and encourage customers to try new brands. The report points out that in the case of issuing coupons, 27.2% customers who originally planned to buy other brands will decide to replace the brand temporarily, 64.3% customers who have no specific brand purchase plan will change the brand. There are many types of cou- pons, which are widely applied in practice or discussed among theoretical workers, such as direct mail coupons [1- 3], coupons published in newspaper or magazine [4,5], and package coupons [6].

This study will mainly focus on the Cross-ruff Coupon. Cross-ruff Coupon refers to this kind of coupons, which consumers get another assigned product or brand dis-count coupons during the purchase process of a particular brand or product, resulting in the purchase of the assig- ned product/brand can get a discount. For example, a cus- tomer enters a beverage store to buy a bottle of Diet Co- ke (a Coca Cola Company’s low-calorie cola soft drinks), at the same time, he received a Diet Sprite (a Coca Cola Company’s low calorie lemon flavored soft drinks) dis- count coupons. This coupon gives an advantage in the lin-

kage between different brands/products consumers pur- chase activity. During September 28, 2008 to January 3, 2009, HP Company has provided coupons of type DJ D1560, DJ D1455 and DJ D4360 printers for purchasers who buy any type of HP notebook or desktop computers.

In order to the convenience of description, we define brand/product which issues coupons as Carrier Brand, and define promotion brand/product as Target Brand. According to the survey, cross coupon acceptance ratio is the double of coupons published in newspaper. The aca- demic community’ study on coupons mainly focuses on the influence on the choice of general coupons [2,6,7], in- fluence on consumer behavior [8], the influence on social welfare [9] etc. However, cross coupons is different from direct mail coupons, coupons published in newspaper or magazine, consumers buy other products when they acc- ept cross coupons, so it links different buy behavior. Cro- ss coupons is also different from package coupons, the la- ter can only make consumers buy the same product, cross coupons achieve the aim that promote another product (i.e. target brand) by purchasing one brand product (i.e.

stions as the optimal coupon value, and how to determine the price of products of carrier brand and target brands respectively. Therefore, this paper will focus on resolv- ing joint decision of pricing and cross-ruff coupon value under substitute (complement) demand conditions based on the literatures.

2. Analysis for Decision Model

Raju [6] proposed that there are different distribution forms when enterprise issues coupons with product pack- age, according to cost, product characteristics, policy con- straints and other actual conditions, including In-Pack coupons, which is packaged within the package, and On- Pack coupons, which is printed or pasted on the outer pac- kage. Accordingly, this paper will mainly constructs two kinds of models, i.e. optimal decision model for cross-

ruff coupons within the package and cross-ruff coupons printed or pasted on the package. In order to the conci- sion for writing, the following use “IPC coupons” instead of coupons which is packaged within the package, and use “OPC coupons” instead of coupons which is printed or pasted on the outer package. The desired symbol is de- fined as follows:

D = Demand for product P = Price of product unit R = Value of Cross-ruff Coupon C = Cost of product unit

w = Cost of acceptance of coupon unit

π= Profit of enterprise

In the process of building model, we use subscript c

represent carrier brand, use t represent target brand, use s

represent demand condition, use m represent the demand

is complementary, use d represent the demand is inde-

pendent. And as for all symbols “±” or “ ”, the upper part represents demand for substitutive conditions, the lower part represents demand for complementary condi-tions, such as “±” represent that when the demand for carrier brand and target brand demand is substitutive, we take “+”; when the demand for them is complementary, we take “–”. Firstly, we develop and analyze the IPC coupons optimal value decision model, and then, we give the same process of OPC coupons.



2.1. IPC Coupons

Assume that the demand is linear function, due to the substitutive (complementary) relationship between two

products demand, the change of one product’s a price will affect the demand for another product. At the same time, according to the definition of IPC coupon, we know that at the time of purchase, customers do not know whe- ther there are coupons in the package, only when the pur- chase completed and the package is opened, they find coupons and determine whether exchange. So this kind of coupons has no effect on the needs of carrier brand no effect, it can only affects the demand for target brand. Therefore, the demand function of carrier brand and tar- get brand can be expressed as follows respectively.

c c c c c t

D a b P c P, _{t t t t t t c}

Among them, Dc represents the demand of carrier

brand, Dt represents the demand of target brand, ac, bc, c

D a b Pc PrD

c,

at, bt, ct are parameters of demand function of carrier

brand and target brand. r represents the acceptance rate

of coupons, it is determined by the relative size of cou-pon value and reference price P0, and P0 is the maximum value of coupons when there is 100% coupon acceptance ratio, function form as follows [11]:

0

R r

P 

Profit function of enterprise can be represented as fol-lows:











πDc PcCc D Pt tCt rDc Rw



(1)

The first order partial derivative of profit function with respect to variable R can be represented as follows:





0

π c c c c t ₂

t t

a b P c P

P C R w

R P

 

 _{  }

  (2)

Appoint Equation (2) is equal to zero, we can get

2

t t

P C w

R   (3)

Take Equation (3) into Equation (1), the first order partial derivative of profit function with respect to Pc, Pt

can be represented as follows respectively:









2

0

π

2

4

c t t c c c c t t c c t t

c

b P C w

a b P c c P b C c C

P P

 

 _{     }

 

(4)















2

0

0

π 2

4

2

c t t t t t c t c t t c c

t

c c c c t t t

c P C w

a b P c c P b C c C

P P

a b P c P P C w

P

 

 _{     }



   





(5) Further get Hessian Matrices of profit function to Pc and Pt:











0



0 0

2

2

2

2 2

c t t

c c

c t t c t t c c c c t

c t t

b P C w

b c

P H

b P C w c P C w a b P c P

c c b

P P

 

 

   

 

 

  _{     } 

      

 

 0 

t

c

P

The first and second order principle which can judge the characteristic of H is D1 and D2 respectively, andD1  2bc,





2

2

0

4 4 c c

c t c c c t c

b D

D b b rb c rb c c

P

      .

As a concave function, profit function π ask D₁0,

and D₂ 0, if





2

0

4 4 c c

c t c c c t c

b D

b b rb c rb c c

P

  

   0

Simplified it, condition (7) can be obtained



2

0

4 4 c c

c t c c c t c

b D

b b rb c rb c c

P

 

  



(7)

Because

0

1 2

t t

P C w

r

P

 

  , by a validated numerical experiments indication，condition (7) is true at actual si- tuation of enterprise operation.

Let (4), (5) are equal to zero, dissolve the equation sys- tem, we can get:

1) Under the substitutive conditions, the optimal prod- uct price of carrier brand and target brand and

are ipc cs P ipc ts P



 



2

0

2 8

ipc ipc ipc

c c c c ts t ts t ts t ipc

cs

c

a b C c P c P C P C w

P b P         (8)















2 0 0 2 8 4

ipc ipc ipc

t t t t cs c cs c c ts t ipc

ts

t

ipc ipc ipc c c cs c ts ts t

t

a b C c P c P C c P C w

P

b b

a b P c P P C w

b P              tP (9) 2) Under the complementary conditions, the optimal product price of carrier brand and target brand and

are ipc cm P ipc tm P



 



2

0

2 8

ipc ipc ipc

c c c c tm t tm t tm t ipc

cm

c

a b C c P c P C P C w

P b P         (10)















2 0

0 2 4 8 ipc ipc t t t t cm c cm c ipc

tm

t

ipc ipc ipc c c cm c tm tm t

t ipc

c tm t

t

a b C c P c P C P

b

a b P c P P C w

b P

c P C w

b P              (11)

Let , , the optimal product price of car- rier brand and target brand under independent condition

and are 0 c c  ipc td P 0 t c  ipc cd P





2

0

2 8

ipc td t ipc c c c

cd

c

P C w

a b C P

b P

 



  (12)







0

2 4

ipc ipc c c cd td t ipc t t t

td

t t

a b P P C w

a b C P

b b P

 



   (13)

Take the optimal price of target brand into (3), the optimal value of IPC coupons can be obtained.

ipc t P_ ipc R_

Compare expressions (8)-(13), such conclusion can be educed:

Conclusion 1: When IPC coupons are issued，the opti- mal price of carrier brand and target brand, the value of coupons is higher at the condition of substitutive than that of complementary, that is, ipc ipc ipc

s d

P_ P_ Pm , and ipc ipc ipc

s d m

R R R .

2.2. OPC Coupons

According to the definition of OPC coupons, customers have known that there are coupons when they buy one product. So this kind of coupons can not only excite cus-tomers to buy the carrier brand/product, but also inspire them to buy the target brand/product, namely, there are influence on the demand for the two type of brands. The demand function of carrier brand and target brand can be represented as follows respectively.

c c c c c t c

D a b P c Pd R, D_t a_t b P_{t t}c P_{t c}rD_c

The profit function of enterprise is











πDc PcCc D Pt tCt rDc Rw



(14)

Further changes to



 













0 0 π

c c c c t c c c t t

t t t t c t t

a b P c P d R P P C R P C R w

P

a b P c P P C

   _      _



   

(15) Lemma: the enterprise profit is optimal, if and only if

0

2 2

t t c

c P C w d P R

b

 

  (16)

The prove method refer Khouja [11].

Take (16) into profit function (15), the first order par- tial derivative of profit function with respect to variable

Pc and Pt, we can get





0 2 2 0 0 π 2 2 2 2 2

c c c c c c t t t

c

t t c

c

c

c t t c

c

a b P b C c c P c C P

P C w d P d

b

b P C w d P













2 2 0 0 0 0 π 1 2

2 2 2

2 2 2

c c t t

t t t t t c t c c c c c c

t c

t t t t c

c c c c t c

c

d d P C w d

a b P b C c c P c C P C c

P P

P C w P C w d P

a b P c P d

P b 2 cP b  _{   }   _{       }  _  _  _           _    _                       (18)

Further get Hessian Matrices of profit function to Pc and Pt :









0

0 0

2

2 2

2

2 2 2 2

c t t c

c c

c t t

c c c

c t t c

b P C w

d

b c

P H

b P C w

d D d

c c b c

P P              _{      }       _  _{ }    

 0 

t

t t

c

P C w

P 

(19)

The first and second order principle which can judge the characteristic of H is D1 and D2 respectively, and

, 1 2 c

D   b

tion.

Let (17) (18) are equal to zero, dissolve the equation systems, we can get

1) Under the substitutive conditions, the optimal product price of carrier brand and target brand and

can be represented as follows:

opc cs P opc ts P











2 2 0 4 2 2

c t c c t c

c c

c c c c

D b b d c c b r

b D b r d d c

P

    

    .

when ₁ and ₂ , profit function π is concave function. So, profit function is concave function, if

0

D  D 0



 

2





0

4 2 c c

c t c c t c c c c c

b D b b d c c b r b r d d c

P

      2 

(20)





0 2 2 0 0 2

2 2 2

1

2 2 2

opc opc c c c c ts t ts t opc

cs

c opc

c ts t c

c c

opc

ts t c

c a b C c P c P C P

b

d P C w d P

b b

P C w d P

P b           _  _   _{     }     _{ } _{ }        (21)

By a validated numerical experiments indication, con- dition (20) is true at actual situation of enterprise opera-





_

_

2 2

0

0

0

0

1

2 4 2 2 2

1

2 2 2 2

opc opc _opc

t t t t cs c cs c

opc c opc c ts t

ts cs c c

t t t

opc opc

opc opc

ts t ts t c

c c cs c ts c

t c

a b C c P c P C _{d d P C w}

P P C c

b b b P

P C w P C w d P

a b P c P d

b P b

2 c c d P b                     _  _{  } _{ }    _    _                          _ (22)

2) Under the complementary conditions, the optimal product price of carrier brand and target brand and

can be represented as follows:

opc cm P opc tm P





2 2

0 0

0 1

2 2 2 2 2 2

opc opc _{opc opc}

c c c c tm t tm t

opc c tm t c tm t c

cm

c c c

a b C c P c P C _{d P C w d P P C w d P}

P

b b b P 2bc

 

             

 

  _  _ _{ } _{ }

   

  _  _ (23)





_

_

2 2

0

0

0

0

1

2 4 2 2 2

1

2 2 2 2

opc opc _opc

t t t t cm c cm c

opc c opc c tm t

tm cm c c

t t t

opc opc

opc opc

tm t tm t c

c c cm c tm c

t c

a b C c P c P C _{d d P C w}

P P C c

b b b P

P C w P C w d P

a b P c P d

b P b

2 c c d P b       _{       }      _  _{  } _{ }    _    _                           (24)

Let Cc, Ct, the optimal product price of carrier brand

and target brand under independent condition opc cd

P and

are opc td P 2 2 0 0 1

2 2 2 2 2 2 2

opc opc

opc c c c c td t c td t c

cd

c c c c

a b C d P C w d P P C w d P

P

b b b P b

0                  _  _ _{ } _{ }    





2 2 0 0 0 0 2 4

4 2 2

1

2 2

2 2

opc t t t c opc

td cd c

t t

opc

c td t c

t c

opc

opc td t

c c cd t

opc

td t c

c

c a b C d

P P C

b b

d P C w d P

b P b

P C w

a b P

b P

P C w d P d b                 _{ } _{ }             _{  }                 (26)

Take the optimal price of target brand into (16), the optimal value of OPC coupons can be obtain- ed. opc t P_ opc R_

Compare expressions (21)-(26), such conclusion can be educed:

Conclusion 2: When OPC coupons are issued, the op- timal price of carrier brand and target brand, the value of coupons is higher at the condition of substitutive than that of complementary, that is,

opc opc opc

s d m

P P P ,

opc opc opc

s d m

R R R .

If there is no coupon to issue, the optimal price of two kinds of products can be represented as follows respec-tively





2

no c c c c t t t t no

c

c

a b C c c P c C

P b        (27)





2 no t t t c t c c c no

t

t

a b C c c P c C

P b        (28)

Next, we can further analyze the concrete decision process for IPC and OPC coupons by numerical exam- ples, and validate the conclusions educed above.

3. Numerical Examples

Assume that the demand function of carrier brand is , demand function of target brand is

350, 000 5000 3000

c c

D   P  P_t d R_c



0



350, 000 7000 3000

t t c

D   P P  R P D_c.

Unit cost of carrier brand , Unit cost of target brand , cost of acceptance of coupon unit

, and the ratio of customer exchange coupons is a linear function of the value of coupons, reference price 0 . When enterprise does not issue coupons, namely , the optimal value of two brands are

10 c C  10 t C  0.25 0 R w 90 P 





72 31.92; 40

no no no

cs cm cd

P  P  P 

59 18.46; 30

no no no

ts tm td

P  P  P

no

and

,







the maximum enterprise profit πs 17,852,000



π no m 

; .If enterprise issue cou-

pons, but coupons does not affect the demand of carrier brand, namely IPC coupons, , then the optimal

price of two brands are ;

and ;

4,017,300 πno 7,300,000



d  0 c d  71.47 ipc cs P 

64.18



ipc tm P 



ipc 31.43

cm P  19.10 ipc td P



39.39 ipc cd

P  ipc

ts

P  

, the optimal value of coupons is



31.28 ipc 26.97

s R 



ipc 4.43

m R 

πipc s 

; , the maximum enterprise

profits





πipc 4,044 ;

m  10.51  519,000 ipc d R

21, ,600 ipc

d

 

. Conspicuously, the profit is bigger when enterprise issue coupons than that of don’t issue. When coupons has effect on the demand of carrier brand, namely OPC coupons, let , then the optimal

price of two brands are ;

and ;

，the optimal value of coupons is



7,300,000



50.69 opc cd P 



35.10 opc td P  87.32 opc s R  2500 c d  117.29 opc cs P  96.86





39.19

opc cm P 



opc 18.94

tm P  opc

ts P



opc 22.9₄ opc _34.9 d

R 

πopc 31,113

s 

m ; , the

maxi-mum enterprise profits

R 2



000

,



πopc

m 

; , Conspicuously, the

profit is bigger when enterprise issue coupons than that of don’t issue, and is higher than that of issuing IPC coupons, the above rate is about 44.58%.

4,843, 400 πopc 500



d 9,380,

From the research results, we can see when enterprises use cross ruff coupons for promotions, the stronger the substitution between the target brand and carrier brand, the bigger the enterprise profit is, and the promotion ef-fect is better. Conversely, the stronger the complemen-tary the selected brand, the smaller the enterprise profit increment is, and the promotional effect is worse.

4. Conclusions

According to the enterprise management reality, i.e. dur-

ing the sales process of best-selling products or mature product, enterprises often choose to release coupons to promote the unsalable goods or new product, this paper studied the determination problems of coupons value and product price. Through the construction of joint decision model for two most common cross-ruff coupons (IPC and OPC coupons) value and product price, analyzed the optimal coupon value and the optimal carrier brand and target brand product pricing strategies under different demand structure (substitutive, complementary, indepen- dent). According to the related strategy, this paper put forward the management implications for the choice of associated brand when enterprise chooses coupons to sales promotion, i.e. enterprise should choose the brands

pair which have strong demand substitution as carrier brand and target brand as far as possible, while minimize the needs complementary between associated brands.

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